Logarithm of Infinite Product of Real Numbers

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Theorem

Let $(a_n)$ be a sequence of strictly positive real numbers.

Convergence

The following are equivalent:

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$.
The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ converges to $\log a$.


Divergence to zero

The following are equivalent:

  • The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $-\infty$.


Divergence to infinity

The following are equivalent:

  • The series $\displaystyle \sum_{n \mathop = 1}^\infty\log a_n$ diverges to $+\infty$.


Also see